<div class="problemindexholder" problemindex="C">
 <div class="ttypography">
  <div class="problem-statement">
   <div class="header">
    <div class="title">
     C. Sum of Nestings
    </div>
    <div class="time-limit">
     <div class="property-title">
      time limit per test
     </div>
     2 seconds
    </div>
    <div class="memory-limit">
     <div class="property-title">
      memory limit per test
     </div>
     256 megabytes
    </div>
    <div class="input-file">
     <div class="property-title">
      input
     </div>
     standard input
    </div>
    <div class="output-file">
     <div class="property-title">
      output
     </div>
     standard output
    </div>
   </div>
   <div>
    <p>
     Recall that the bracket sequence is considered regular if it is possible to insert symbols '
     <span class="tex-font-style-tt">
      +
     </span>
     ' and '
     <span class="tex-font-style-tt">
      1
     </span>
     ' into it so that the result is a correct arithmetic expression. For example, a sequence "
     <span class="tex-font-style-tt">
      (()())
     </span>
     " is regular, because we can get correct arithmetic expression insering symbols '
     <span class="tex-font-style-tt">
      +
     </span>
     ' and '
     <span class="tex-font-style-tt">
      1
     </span>
     ': "
     <span class="tex-font-style-tt">
      ((1+1)+(1+1))
     </span>
     ". Also the following sequences are regular: "
     <span class="tex-font-style-tt">
      ()()()
     </span>
     ", "
     <span class="tex-font-style-tt">
      (())
     </span>
     " and "
     <span class="tex-font-style-tt">
      ()
     </span>
     ". The following sequences are not regular bracket sequences: "
     <span class="tex-font-style-tt">
      )(
     </span>
     ", "
     <span class="tex-font-style-tt">
      (()
     </span>
     " and "
     <span class="tex-font-style-tt">
      ())(()
     </span>
     ".
    </p>
    <p>
     In this problem you are given two integers
     <span class="tex-span">
      <i>
       n
      </i>
     </span>
     and
     <span class="tex-span">
      <i>
       k
      </i>
     </span>
     . Your task is to construct a regular bracket sequence consisting of round brackets with length
     <span class="tex-span">
      2·
      <i>
       n
      </i>
     </span>
     with total sum of nesting of all opening brackets equals to exactly
     <span class="tex-span">
      <i>
       k
      </i>
     </span>
     . The nesting of a single opening bracket equals to the number of pairs of brackets in which current opening bracket is embedded.
    </p>
    <p>
     For example, in the sequence "
     <span class="tex-font-style-tt">
      ()(())
     </span>
     " the nesting of first opening bracket equals to
     <span class="tex-span">
      0
     </span>
     , the nesting of the second opening bracket equals to
     <span class="tex-span">
      0
     </span>
     and the nesting of the third opening bracket equal to
     <span class="tex-span">
      1
     </span>
     . So the total sum of nestings equals to
     <span class="tex-span">
      1
     </span>
     .
    </p>
   </div>
   <div class="input-specification">
    <div class="section-title">
     Input
    </div>
    <p>
     The first line contains two integers
     <span class="tex-span">
      <i>
       n
      </i>
     </span>
     and
     <span class="tex-span">
      <i>
       k
      </i>
     </span>
     (
     <span class="tex-span">
      1 ≤
      <i>
       n
      </i>
      ≤ 3·10
      <sup class="upper-index">
       5
      </sup>
     </span>
     ,
     <span class="tex-span">
      0 ≤
      <i>
       k
      </i>
      ≤ 10
      <sup class="upper-index">
       18
      </sup>
     </span>
     ) — the number of opening brackets and needed total nesting.
    </p>
   </div>
   <div class="output-specification">
    <div class="section-title">
     Output
    </div>
    <p>
     Print the required regular bracket sequence consisting of round brackets.
    </p>
    <p>
     If there is no solution print "
     <span class="tex-font-style-tt">
      Impossible
     </span>
     " (without quotes).
    </p>
   </div>
   <div class="sample-tests">
    <div class="section-title">
     Examples
    </div>
    <div class="sample-test">
     <div class="input">
      <div class="title">
       Input
      </div>
      <pre>3 1<br/></pre>
     </div>
     <div class="output">
      <div class="title">
       Output
      </div>
      <pre>()(())</pre>
     </div>
     <div class="input">
      <div class="title">
       Input
      </div>
      <pre>4 6<br/></pre>
     </div>
     <div class="output">
      <div class="title">
       Output
      </div>
      <pre>(((())))</pre>
     </div>
     <div class="input">
      <div class="title">
       Input
      </div>
      <pre>2 5<br/></pre>
     </div>
     <div class="output">
      <div class="title">
       Output
      </div>
      <pre>Impossible<br/></pre>
     </div>
    </div>
   </div>
   <div class="note">
    <div class="section-title">
     Note
    </div>
    <p>
     The first example is examined in the statement.
    </p>
    <p>
     In the second example the answer is "
     <span class="tex-font-style-tt">
      (((())))
     </span>
     ". The nesting of the first opening bracket is
     <span class="tex-span">
      0
     </span>
     , the nesting of the second is
     <span class="tex-span">
      1
     </span>
     , the nesting of the third is
     <span class="tex-span">
      2
     </span>
     , the nesting of fourth is
     <span class="tex-span">
      3
     </span>
     . So the total sum of nestings equals to
     <span class="tex-span">
      0 + 1 + 2 + 3 = 6
     </span>
     .
    </p>
    <p>
     In the third it is impossible to construct a regular bracket sequence, because the maximum possible total sum of nestings for two opening brackets equals to
     <span class="tex-span">
      1
     </span>
     . This total sum of nestings is obtained for the sequence "
     <span class="tex-font-style-tt">
      (())
     </span>
     ".
    </p>
   </div>
  </div>
  <p>
  </p>
 </div>
</div>
